Chiral Properties of $(2\!+\!1)$-Flavor QCD in Magnetic Fields at Zero Temperature

TL;DR

This work delivers a continuum-extrapolated lattice study of (2+1)-flavor QCD in background magnetic fields at $T\simeq 0$, computing renormalized chiral condensates, pseudoscalar masses, and decay constants for $\pi$, $K$, and $\eta^0_{s\bar{s}}$ across $eB$ up to $1.22\,\mathrm{GeV}^2$. It reveals magnetic catalysis in the chiral sector, a monotonic decrease of neutral pseudoscalar masses with $eB$, and a non-monotonic, LLL-consistent but ultimately saturating behavior for charged mesons, with valence-dominated magnetic responses and only modest sea-quark effects. The neutral-pion GMOR relation remains almost preserved over the whole range, while the neutral kaon shows a sizable GMOR violation, highlighting SU(3) dynamics in strong fields. These results provide precise benchmarks for chiral-magnetic phenomena and inform effective theories and future studies of magnetized QCD spectra.

Abstract

We present a lattice QCD study of the chiral properties of $(2\!+\!1)$-flavor QCD in background magnetic fields at zero temperature with physical pion masses. Simulations are performed using the highly improved staggered quark (HISQ) action across four different lattice spacings to enable a controlled continuum extrapolation. We compute the renormalized chiral condensates together with pseudoscalar meson masses and decay constants for pions, kaons, and the fictitious $η^0_{s\bar{s}}$ pseudoscalar as functions of the magnetic field strength $eB$ up to $eB\simeq1.2$ $\mathrm{GeV}^2$. The chiral condensates exhibit clear magnetic catalysis, increasing monotonically with the field strength. In the meson sector, neutral pseudoscalar masses decrease steadily with $eB$, whereas charged pseudoscalar masses display a non-monotonic response: they rise at small fields, consistent with the lowest-Landau-level expectation, but then saturate and slightly decrease at larger fields, signaling sizable internal-structure effects. At the same time, neutral pseudoscalar decay constants are strongly enhanced by the magnetic field. To quantify deviations from chiral symmetry relations, we isolate the magnetic-field-induced shift in the Gell-Mann--Oakes--Renner corrections, and find it to remain small for the neutral pion but to become sizable for the neutral kaon. To elucidate the origin of the magnetic response, we separately analyze the sea and valence quark contributions to both neutral and charged meson masses, finding that valence effects dominate at zero temperature. These results provide new insights into the interplay between QCD chiral symmetry breaking and strong magnetic fields.

Figures (18)

• Figure 1: Temporal correlators $G(n_\tau)$ for the neutral pion's $\bar{u} u$ (top) and $\bar{d} d$ (bottom) components at various values of $eB$ measured on the $24^3\times 48$ ensemble.
• Figure 2: Top: Ground-state mass extraction for $\pi^0_u$ on the $24^3\times 48$ ensemble at $eB=0.81~\mathrm{GeV}^2$. Results from correlated fits with $(N_{\rm nosc},N_{\rm osc})=(1,0),(2,0),(3,0)$ are shown as open symbols versus the fit start time $n_{\tau,\min}$. Black stars denote the AICc-selected fits, and the gray band indicates the final plateau value. Masses extracted using the oblique Lanczos method (red crosses) are also plotted against Euclidean time $n_\tau$. Bottom: The corresponding matrix element $C_{PP} V_s$ obtained from the same set of fits; this quantity is not accessible in the Lanczos method.
• Figure 3: Masses of neutral pseudoscalar meson as functions of the background magnetic field $eB$. The top, middle, and bottom panels show results for $\pi^0$, $K^0$, and $\eta^0_{s\bar{s}}$, respectively. Open symbols denote lattice measurements on the individual ensembles, while the red filled symbols show the corresponding continuum extrapolated results. The insets display the meson masses normalized to their $eB=0$ values to highlight the relative changes.
• Figure 4: Masses of charged pseudoscalar mesons as functions of the background magnetic field $eB$. The top and bottom panels show results for $\pi^\pm$ and $K^\pm$, respectively. Open symbols denote lattice QCD results on the individual ensembles, while the red filled symbols show the continuum extrapolated results. In each panel, the dashed line represents the LLL prediction for a point-like charged particle with the corresponding zero-field mass.
• Figure 5: Ratio of the neutral pion mass components, $M_{\pi_u^0}/M_{\pi_d^0}$, as a function of the magnetic field strength $eB$ for four different lattice spacings. Results obtained from the AICc-selected multi-state fits and the oblique Lanczos method are denoted by red squares and blue circles, respectively.